O uso da tecnologia digital, há anos, faz cada vez mais parte do cotidiano dos estudantes e, por isso, a função do professor é cada vez mais desafiadora. A fim de promover aulas mais atrativas, o professor necessita assumir um novo papel. Nos tempos de hoje, o educador precisa proporcionar para seus alunos aulas diversificadas, que façam uso da tecnologia digital de forma eficaz, e apresentem o conteúdo de forma interdisciplinar e/ou contextualizada, de forma a colocar o aluno no centro do processo de ensino-aprendizagem, estimulá-los e desafiá-los. Em meio a esse contexto e com o objetivo de auxiliar os professores na abordagem do tema Funções Trigonométricas, tão presente no ENEM e com aplicações em diversas áreas, como na Física e na Música, a partir do uso de tecnologias digitais, foi utilizado o software Geogebra para construção de gráficos de diversas dessas funções, além da elaboração de listas de exercícios retirados do ENEM. A cada questão do ENEM escolhida, foram criadas perguntas norteadoras, que podem ser respondidas com o auxílio da investigação das alterações sofridas no gráfico de uma função ao alterarmos sua lei de formação, com o auxílio de gráficos criados no Geogebra, que fazem uso de controles deslizantes, e que podem ser acessados a qualquer momento por alunos e professores a partir dos links e do blog que foram disponibilizados, fazendo com que se tenha interação a medida que as questões forem resolvidas. Dessa forma, espera-se que a aula possua um caráter mais investigativo, com o aluno no centro do processo, facilitando seu entendimento e aprendizagem

One of the main causes of the great di culty of learning mathematics is the lack of the interest from the students. Aiming to seek an improvement in teaching in order to make students more motivated, this project was developed, which proposes a di erent methodology in the teaching of trigonometric functions with the utilization of mathematic software Winplot. Activities were developed for application of Winplot which might provide to the teacher a more e cient method in the teaching of trigonometric functions. The use of computers in mathematics education aims to make young people be motivated to learn this discipline, changing the routine of class and arousing the interest of the student involved. The Mathematics has undergone signi cant changes in their teaching methods, thus develop logical reasoning, encourage independent thinking, creativity and problem-solving skills became essential to all citizen who wishes to have an active and critical participation in society. Therefore, the utilization of informatics in teaching mathematics o ers these characteristics in a simple and fun without leaving to encourage the student in their learning.

This dissertation aims to present a teaching proposal that makes it possible to explore and better understand important aspects of the teaching of Trigonometric Functions, using the educational software GeoGebra. The main question is: What potential does this educational technological tool have to contribute to teaching and learning the concepts of Trigonometric Functions? This issue is relevant since several external assessments (Basic Education Assessment System (SAEB), Diagnostic Learning Assessment (ADA), Goiás Education Assessment System (SAEGO)) carried out in High School in the state of Goiás point out that there is, at national and local level, a gap in the teaching and learning of Mathematics. Thus, it is believed that the use of digital technologies can contribute to making Mathematics teaching more attractive to students. This is a research with a qualitative approach, exploratory and bibliographic, in which activities were developed, initially to improve and understand the software and, subsequently, didactic sequences aimed at students in the 2nd grade of High School, with tasks at an increasing level. requirements related to the Sine, Cosine and Tangent Trigonometric Functions. In the elaboration, the dialogue with didactic books and alternatives was sought to dynamically explore the potential of the software, in order to promote the understanding of both the steps and the content covered, through the suggested constructions. It is believed that the ideas presented here may serve as support for various works and projects related to the teacher's own practice or even students of exact science courses who still have difficulties with the subject. With the proposal, it is expected to add more formative possibilities for the teaching of Mathematics and, in particular, for teaching and learning with the help of digital technologies, especially GeoGebra.

In this work the main goal is focused on applying the theory of Taylor polynomial approximations applied on the trigonometric function defined by f : [0; 2 ] ! R, where f(x) = sin(x). To achieve this goal, eight sections were developed, in which initially a reflection on the problem and the need to obtain the values in this respect in that it is wide angle measure x is presented. Is presented and subsequently treated a problem involving the movement of a pendulum, which uses the approximation sin(x) x where x belongs to a certain range. In the sections that follow a literature review of the theories of differential and integral calculus is presented, and the related theory of Taylor approximation of functions by polynomials. Later we used these theories to analyze and determine polynomials approximating the function f(x) = sin(x) in a neighborhood of the point x = 0, and estimate the error when we applied these approaches. At this time the error occurred due to the approach used in the pendulum problem was also analyzed. Finally a hint of practice to be held in the classroom using the theories treated here as well as the study of the problem of heat transfer in a bar through the theory of Fourier activity is presented.

O objetivo desse trabalho é relatar a construção de um material útil no ensino de trigonometria tanto para professores do último ano do segundo segmento do Ensino Fundamental quanto para professores do primeiro ano do Ensino Médio. Aqui procura-se aplicar uma possível fusão entre uma sequência didática experimental e a abordagem histórica no Ensino da Trigonometria, fazendo dessas um possível agente motivador, principalmente àqueles alunos com maior dificuldade de abstração algébrica e àqueles com maior interesse nas áreas humanas. As articulações pedagógicas entre a trigonometria e sua história expostas aqui serão complementadas de ideias práticas para melhor assimilação dos estudantes ao conteúdo em si, assim como a observação de seus resultados

It is presented an alternative formulation to solve the problem of the deformation analysis for tubular element under pinching loads. The solution is based on a new displacement field defined from a total set of trigonometric functions. The solution is developed in a multi-nodal finite tubular ring element with a total of eight degrees of freedom per section considered. The purpose of this paper is to provide an easy alternative formulation when compared with a complex finite shell element or beam element analysis for the same application. Several case studies presented have been compared and discussed with numerical analyses results reported by other authors and the results obtained with a shell element from a Cosmos/M® programme.

It is presented an alternative formulation to solve the problem of the deformation analysis for tubular element under pinching loads. The solution is based on a new displacement field defined from a total set of trigonometric functions. The solution is developed in a multi-nodal finite tubular ring element with a total of eight degrees of freedom per section considered. The purpose of this paper is to provide an easy alternative formulation when compared with a complex finite shell element or beam element analysis for the same application. Several case studies presented have been compared and discussed with numerical analyses results reported by other authors and the results obtained with a shell element from a Cosmos/M® programme.

A displacement formulation for cylindrical shells subjected to radial loads is presented. Static loads are applied to straight thin pipes using a variational method and results are compared both to finite element analysis and a shell element from Ansys program. The formulation includes a combination of trigonometric functions in two variables, longitudinal and circumferential direction. Then a system of differential equations is solved for two different load cases and the vertical displacement is obtained in the points where the load is applied.

The main objective of this work is to present a numerical formulation to solve the problem of the deformation analysis ot thin-walled circular cylindrical pipes under concentred loads. The solutions is based on a displacement field entirely defined from a set of trigonometic functiobns where the amplitudes are assigned as a nodal paramenters in a multi-nodal finite element. With this formulation it is possible to provide an easy alternative tool when compared with a complex finite shell or solid element modelling for the same type of applications. The present work permits to examine the defection of pipe rings subjected to lateral (transverse) static loading conditions. Several case studies presented have been compared and discussed with numericalç analysisresults obtained with a shell element from Ansys programme.